Problem 17E
Let \(X\) ∼ Geom(p), let \(n\) be a non-negative integer, and let \(Y\) ∼ Bin(n, p). Show that P(X = n) = (1/n)P(Y = 1).
Equation Transcription:
Text Transcription:
X
n
Y
Step-by-Step Solution:
Solution :
Step 1 of 1:
Given and
Where mean and variance
.
Here non-negative integer is n.
Our goal is :
We need to prove that P(X=n)=.
Now we have to prove that P(X=n)=.
P(X=n)=
The formula of the geometric distribution is
P(X=n)=
Where q = (1-p)
So P(X=n)=
Then,
P(X=n)=
The formula of the binomial distribution is
Here the probability of X is n and the probability of Y is 1.
P(X=n)=
Here and
= p.
P(X=n)=
Therefore P(X=n)=.
Step 2 of 1
